Integral of cos(x) exp(-x^2)

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Integral of cos(x) exp(-x^2)

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Learn more \bold{\mathrm{Basic}} \bold{\alpha\beta\gamma} \bold{\mathrm{AB\Gamma}} \bold{\sin\cos} \bold{\ge\div\rightarrow} \bold{\overline{x}\space\mathbb{C}\forall} \bold{\sum\space\int\space\product} \bold{\begin{pmatrix}\square&\square\\\square&\square\end{pmatrix}} \bold{H_{2}O} \square^{2} x^{\square} \sqrt{\square} throot[\msquare]{\square} \frac{\msquare}{\msquare} \log_{\msquare} \pi \theta \infty \int \frac{d}{dx} \ge \le \cdot \div x^{\circ} (\square) |\square| (f\:\circ\:g) f(x) \ln e^{\square} \left(\square\right)^{'} \frac{\partial}{\partial x} \int_{\msquare}^{\msquare} \lim \sum \sin \cos \tan \cot \csc \sec \alpha \beta \gamma \delta \zeta \eta \theta \iota \kappa \lambda \mu u \xi \pi \rho \sigma \tau \upsilon \phi \chi \psi \omega A B \Gamma \Delta E Z H \Theta K \Lambda M N \Xi \Pi P \Sigma T \Upsilon \Phi X \Psi \Omega \sin \cos \tan \cot \sec \csc \sinh \cosh \tanh \coth \sech \arcsin \arccos \arctan \arccot \arcsec \arccsc \arcsinh \arccosh \arctanh \arccoth \arcsech \begin{cases}\square\\\square\end{cases} \begin{cases}\square\\\square\\\square\end{cases} = e \div \cdot \times < > \le \ge (\square) [\square] \:\longdivision{} \times \twostack{}{} + \twostack{}{} - \twostack{}{} \square! x^{\circ} \rightarrow \lfloor\square\rfloor \lceil\square\rceil \overline{\square} \vec{\square} \in \forall otin \exist \mathbb{R} \mathbb{C} \mathbb{N} \mathbb{Z} \emptyset \vee \wedge eg \oplus \cap \cup \square^{c} \subset \subsete \superset \supersete \int \int\int \int\int\int \int_{\square}^{\square} \int_{\square}^{\square}\int_{\square}^{\square} \int_{\square}^{\square}\int_{\square}^{\square}\int_{\square}^{\square} \sum \prod \lim \lim _{x\to \infty } \lim _{x\to 0+} \lim _{x\to 0-} \frac{d}{dx} \frac{d^2}{dx^2} \left(\square\right)^{'} \left(\square\right)^{''} \frac{\partial}{\partial x} (2\times2) (2\times3) (3\times3) (3\times2) (4\times2) (4\times3) (4\times4) (3\times4) (2\times4) (5\times5) (1\times2) (1\times3) (1\times4) (1\times5) (1\times6) (2\times1) (3\times1) (4\times1) (5\times1) (6\times1) (7\times1) \mathrm{Radians} \mathrm{Degrees} \square! ( ) % \mathrm{clear} \arcsin \sin \sqrt{\square} 7 8 9 \div \arccos \cos \ln 4 5 6 \times \arctan \tan \log 1 2 3 - \pi e x^{\square} 0 . \bold{=} + \mathrm{simplify} \mathrm{solve\:for} \mathrm{inverse} \mathrm{tangent} \mathrm{line} See All area asymptotes critical points derivative domain eigenvalues eigenvectors expand extreme points factor implicit derivative inflection points intercepts inverse laplace inverse laplace partial fractions range slope simplify solve for tangent taylor vertex geometric test alternating test telescoping test pseries test root test Related ? Graph ? Number Line ? Similar ? Examples ? Our online expert tutors can answer this problem Get step-by-step solutions from expert tutors as fast as 15-30 minutes. Your first 5 questions are on us! In partnership with You are being redirected to Course Hero I want to submit the same problem to Course Hero Examples step-by-step \int\cos(x^{2})dx en Feedback Something went wrong. Wait a moment and try again. Method 1 (Contour integration): $$f(x)=e^{-x^2}$$ Let $C$ be a contour that is a rectangle with vertices at $-R$,$R$, $R+i/2$ and $-R+i/2$. Letting $R\to\infty$, the integral along the sides disappears, so by Cauchy's Integral Theorem: $$ 0 = \oint_C f(z) = \\ \int_{-\infty}^\infty f(x)\, dx-\int_{-\infty}^\infty f(x+i/2)\, dx = \\ \sqrt{\pi}-e^{1/4}\int_{-\infty}^\infty e^{-x^2}(\cos(x)+i\sin(x))\, dx $$ Taking the real parts of both sizes, we obtain $$\int_{-\infty}^\infty e^{-x^2}\cos(x)\,dx = \frac{\sqrt \pi}{\sqrt[4] e}$$ Method 2 (Differentiating under the Integral Sign): $$I(a) = \int_{-\infty}^\infty e^{-x^2} \cos(a x)\,dx$$ $$I'(a) = -\int_{-\infty}^\infty x e^{-x^2} \sin(a x)\,dx = \frac{1}{2}e^{-x^2} \sin(a x)\bigg|_{-\infty}^\infty-\frac{a}{2}\int_{-\infty}^\infty e^{-x^2}\cos(ax)\,dx = -\frac{a I(a)}{2}$$ Because $I(0)=\sqrt{\pi}$ we have $I(a) = \sqrt \pi e^{-\frac{a^2}{4}}$. Then $I(1) = \frac{\sqrt \pi}{\sqrt[4] e}$ is the answer. Method 3 (Summation): $$\cos x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}$$ so $$ I = \int_{-\infty}^\infty e^{-x^2}\cos x\,dx = \\ \int_{-\infty}^\infty e^{-x^2}\sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}\, dx=\\ \sum_{n=0}^\infty \frac{(-1)^n} {(2n)!}\int_{-\infty}^\infty e^{-x^2} x^{2n}\, dx=\\ \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}2\int_{0}^\infty e^{-x^2} x^{2n}\, dx \stackrel{x\mapsto \sqrt x}{=}\\ \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}\int_{0}^\infty e^{-x} x^{n-1/2}\, dx =\\ \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}\Gamma\left(k+\frac{1}{2}\right)=\\ \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} \frac{\sqrt{\pi} (2n)!}{4^n n!} = \\ \sqrt \pi \sum_{n=0}^\infty \left(-\frac{1}{4}\right)^n \frac{1}{n!} = \\ \frac{\sqrt \pi}{\sqrt[4] e} $$ Where this is employed and the change of summation and integration must be justified. This website uses cookies to ensure you get the best experience. By using this website, you agree to our Cookie Policy. Learn more \bold{\mathrm{Basic}} \bold{\alpha\beta\gamma} \bold{\mathrm{AB\Gamma}} \bold{\sin\cos} \bold{\ge\div\rightarrow} \bold{\overline{x}\space\mathbb{C}\forall} \bold{\sum\space\int\space\product} \bold{\begin{pmatrix}\square&\square\\\square&\square\end{pmatrix}} \bold{H_{2}O} \square^{2} x^{\square} \sqrt{\square} throot[\msquare]{\square} \frac{\msquare}{\msquare} \log_{\msquare} \pi \theta \infty \int \frac{d}{dx} \ge \le \cdot \div x^{\circ} (\square) |\square| (f\:\circ\:g) f(x) \ln e^{\square} \left(\square\right)^{'} \frac{\partial}{\partial x} \int_{\msquare}^{\msquare} \lim \sum \sin \cos \tan \cot \csc \sec \alpha \beta \gamma \delta \zeta \eta \theta \iota \kappa \lambda \mu u \xi \pi \rho \sigma \tau \upsilon \phi \chi \psi \omega A B \Gamma \Delta E Z H \Theta K \Lambda M N \Xi \Pi P \Sigma T \Upsilon \Phi X \Psi \Omega \sin \cos \tan \cot \sec \csc \sinh \cosh \tanh \coth \sech \arcsin \arccos \arctan \arccot \arcsec \arccsc \arcsinh \arccosh \arctanh \arccoth \arcsech \begin{cases}\square\\\square\end{cases} \begin{cases}\square\\\square\\\square\end{cases} = e \div \cdot \times < > \le \ge (\square) [\square] \:\longdivision{} \times \twostack{}{} + \twostack{}{} - \twostack{}{} \square! x^{\circ} \rightarrow \lfloor\square\rfloor \lceil\square\rceil \overline{\square} \vec{\square} \in \forall otin \exist \mathbb{R} \mathbb{C} \mathbb{N} \mathbb{Z} \emptyset \vee \wedge eg \oplus \cap \cup \square^{c} \subset \subsete \superset \supersete \int \int\int \int\int\int \int_{\square}^{\square} \int_{\square}^{\square}\int_{\square}^{\square} \int_{\square}^{\square}\int_{\square}^{\square}\int_{\square}^{\square} \sum \prod \lim \lim _{x\to \infty } \lim _{x\to 0+} \lim _{x\to 0-} \frac{d}{dx} \frac{d^2}{dx^2} \left(\square\right)^{'} \left(\square\right)^{''} \frac{\partial}{\partial x} (2\times2) (2\times3) (3\times3) (3\times2) (4\times2) (4\times3) (4\times4) (3\times4) (2\times4) (5\times5) (1\times2) (1\times3) (1\times4) (1\times5) (1\times6) (2\times1) (3\times1) (4\times1) (5\times1) (6\times1) (7\times1) \mathrm{Radians} \mathrm{Degrees} \square! ( ) % \mathrm{clear} \arcsin \sin \sqrt{\square} 7 8 9 \div \arccos \cos \ln 4 5 6 \times \arctan \tan \log 1 2 3 - \pi e x^{\square} 0 . \bold{=} + \mathrm{simplify} \mathrm{solve\:for} \mathrm{inverse} \mathrm{tangent} \mathrm{line} See All area asymptotes critical points derivative domain eigenvalues eigenvectors expand extreme points factor implicit derivative inflection points intercepts inverse laplace inverse laplace partial fractions range slope simplify solve for tangent taylor vertex geometric test alternating test telescoping test pseries test root test Related ? Graph ? Number Line ? Similar ? Examples ? Our online expert tutors can answer this problem Get step-by-step solutions from expert tutors as fast as 15-30 minutes. Your first 5 questions are on us! In partnership with You are being redirected to Course Hero I want to submit the same problem to Course Hero Examples step-by-step \int e^x\cos (x)dx en Feedback Yusuke Kawasaki/Flickr Xacuti, xiaolongbao, ximenia, xoconostle and xpinec are just some of the foods that begin with the letter "X." Because so few words begin with the letter "X" in English, all of these foods come from countries outside the United States. Xacuti This brown curry comes from the village of Arambol in the Indian region of Goa and is usually made with chicken or seafood. Xacuti is often served with rice, bread or even over an omelet. It can vary in spiciness. Ingredients include pepper, onion, white poppy seeds, fresh and dried chilies, turmeric, cinnamon, cloves, nutmeg and other spices. Xiao Long Bao A small (xiao) basket (long) bun (bao) is a steamed dumpling filled with broth and pork. Originally from Shanghai, China, xiao long baos can now sometimes be found in American Chinese restaurants in big cities like Chicago. Eating these dumplings can be difficult for people who haven't tried them before. They are served in a bamboo steamer while extremely hot, making it risky to eat one too soon. However, the longer you wait for the xiao long bao to cool, the more likely it is that the bottom will tear, losing the delicious broth. To overcome these challenges, pick the dumpling up with chopsticks and place it on a soup spoon. (You may wish to add vinegar and ginger to the spoon beforehand for added flavor.) As the dumpling cools on the spoon, consider piercing the skin of the dumpling with a fork or your teeth to help it reach a safe temperature faster. Once a few minutes have gone by and the dumpling is ready, slide the dumpling (and any vinegar and ginger) into your mouth and enjoy. Ximenia Ximenia is the name of both a tree and its fruit that grows in countries such as Ethiopia, Tanzania and South Africa. Its English name comes from Francisco Ximenez, a Spanish monk. The fruit is orange or red with white spots and only slightly longer than an inch. The taste is bitter and tart. The skin should be peeled and discarded before eating, although the nut is edible. Ximenia fruit are used for jams, jellies, deserts and as a sweetener for porridge. They are also eaten raw. Additionally, the roots and leaves of the tree can be used for medicinal purposes, such as treating fevers or inflamed eyes. Xnipec The name of this fiery salsa comes from the Mayan words for "dog's nose" or "dog's snout," probably because the heat of it will make your nose run wet like a dog's. Xnipec is originally from the Yucatan Peninsula, the southeastern part of Mexico where Mayans still live today, but it has since made its way north of the border. The spiciness of this salsa comes from habanero peppers, so be careful when preparing and eating xnipec. Xonocostle Like Ximenia, Xonocostle is a fruit. It comes from a cactus that goes by the same name and grows in Central Mexico. It's used in marinades, salsas, mole de olla and even beverages. Sometimes it's also dried or candied before being eaten. The cactus itself is pale green, while the fruit is deep red and grows at the end of paddle-shaped growths. The taste of xonocostle is sour and acidic. MORE FROM

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